Econ 520, Final Review QuestionsNote: Many of these questions are drawn from previous years’ finals. I will not provide solutionsto these questions. The final will cover material from the entire semester, but with more weighton the second half of the course.1. Let X1,..., X10be a random sample of size 10 from a normal distribution with mean zero(known), and variance σ2. We want to test the null hypothesis H0: σ2= 1 against thealternative hypothesis Ha: σ2>1. Consider a test that rejects the null hypothesis ifPix2i>C, where C is a constant. Find the constant C so that the test has significance level .05.(Hint: use a table of the chi-square distribution.)2. Let the random variable X have probability density function fX(x;θ) =(1/θ) exp(−x/θ) forx >0. Consider the simple hypothesis H0: θ =2 and the alternative hypothesis Ha: θ =4.Let X1, X2denote a random sample of size two from this distribution. Show that the besttest of H0against Hamay be carried out by use of the statistic X1+X2. Find the optimalcritical region for α =0.1.3. Suppose that Xifor i =1,...,n are i.i.d. withPr (Xi=0) =γ,Pr (Xi=1) =γ(1 −γ)Pr (Xi=2) =1 −2γ +γ2,where the parameter is γ ∈(0,1).(a) Verify that the single-observation score function has mean zero, and determine theCramer-Rao bound for unbiased estimators of γ based on (X1,..., Xn).(b) Construct a two-dimensional sufficient statistic for γ based on (X1,..., Xn).(c) Suppose you are given that n =100, the MLEˆγ =.45 and its standard error is SE =.03.For the hypothesis that γ =.5 versus the two-sided alternative, carry out a Wald test.(d) Suppose Yi= 1 if Xi= 0, and Yi= 0 otherwise. Provide an estimator of γ based onlyon the Y1,...,Yn, and calculate its mean squared error.(e) Using the estimator you propose in part (d), show how to construct an approximate95% confidence interval for γ.4. Suppose that X1is uniformly distributed on (0,1) and X2is uniformly distributed on (0,2),and X1and X2are independent. Let Y ≡max(X1, X2).(a) Derive the CDF and PDF of Y .1(b) Calculate E[Y ].(c) Suppose that instead of being independent, X1and X2are related in the followingway: X1is uniform (0,1), and X2= 2 ·X1. Let Y be defined as before. Now calculatethe CDF and PDF of Y .5. Suppose Xn∼Bin(n,αn), where α >0.(a) Xnd→Y . What’s the probability mass function for Y ?(b) DoesXnpnp→0? Prove your answer.6. Suppose that Y1,...,Ynare IID with a discrete distribution with PMFfY(y;K ) =(1Kfor y =1,2,...,K0 otherwise.The parameter K is an integer ≥1.(a) Find the maximum likelihood estimator for the parameter K .(b) Show that if K0is the true value of the parameter, then the MLEˆK has the followingproperties:Pr (ˆK =1) >0;Pr (ˆK >K0) =0.(c) Use your result from (b) to show that the MLEˆK is biased towards 0.7. Let X be a random variable with probability density functionfX(x;µ) =1µexp(−x/µ),for x >0 and zero elsewhere.(a) Calculate the mean and variance of X .(b) Calculate the mean and variance of X conditional on X < 8.(c) Let x1, x2,..., xNbe a random sample from this distribution, with N = 20,Px = 95,andPx2=590. Calculate the maximum likelihood estimate.(d) Test the hypothesis that µ =4 at the 10% level using a likelihood ratio test.(e) Test the same hypothesis using a Lagrange multiplier (score) test.8. Let the marginal distribution of X be binomial with N = 1 and p = 1/4. Conditional on X ,the random variable Y has a normal distribution with mean µ ·(X +1) and variance 1.2(a) Find the marginal density of Y .(b) Suppose you have a random sample of size N from this joint distribution. What is themaximum likelihood estimator and its large sample variance?(c) Suppose you only observe y1,..., yN. Find an unbiased estimator for µ. What is itslarge sample variance and how does that compare to that of the maximum likelihoodestimator derived before?9. Suppose there is a random sample of size 10 (X1,..., X10) from a Poisson distribution (sofXi(xi|θ) =θxie−θxi!, when xiis a nonnegative integer). The sample mean of the randomsample is 4.(a) Using the sample, what is the maximum likelihood estimate for θ?(b) Provide an approximate 95% confidence interval for θ.(c) Using a large-sample LR test, test the hypothesis that θ =3 at the 0.05 level. (Note thatif Z is a chi-squared random variable with 1 degree of freedom, Pr (Z >3.84) =0.05.)(d) Now suppose that the prior distribution for θ is Gamma(3,5):θ ∼G amma(3, 5).What is the posterior distribution for θ given X1,..., X10? (Hint: Z ∼Gamma(α,β)means fZ(z|α,β) =zα−1e−z/βΓ(α)βα)10. A firm samples machine parts until it finds a defective part; let Xibe the number of sam-ples until a defective part. Assume that each sample is independent with probability p ofbeing defective.(a) Derive the probability mass function of Xi.(b) Suppose that the firm obtains independent observations X1,..., Xn. The firm has aBeta(1,1) prior distribution for p. What is the posterior distribution for p?Note: the Beta distribution with parameters α and β has PDFf (x) =Γ(α +β)Γ(α)Γ(β)·xα−1(1 −x)β−1.11. Suppose we have a random sample of individuals from a population, and assume thatearnings in a given month for the ith sampled individual is Xi, where Xihas PDFf (x;θ) =1xp2πexpµ−(log x −θ)22¶, x >0,where the parameter θ ∈R.3(a) Is the statistical model an exponential family?(b) Derive the MLE for θ based on a sample of size n. Is the MLE a minimum varianceunbiased estimator?(c) Construct a large sample Wald test for the hypothesis that θ = θ0. (Hint: if possible,show that the single-observation score function has variance 1. If you cannot showthis, you can just take as given that its variance is 1.)(d) Suppose in our data set we observe n =100,PiXi=538,Pilog Xi=127, andPiX2i=5628. Calculate the MLE and provide a large sample 95% confidence interval for